Boundary Element Model for Nonlinear Fractional-Order Heat Transfer in Magneto-Thermoelastic FGA Structures Involving Three Temperatures

Abstract

The principal objective of this chapter is to introduce a new fractional-order theory for functionally graded anisotropic (FGA) structures. This theory called nonlinear uncoupled magneto-thermoelasticity theory involving three temperatures. Because of strong nonlinearity, it is very difficult to solve the problems related to this theory analytically. Therefore, it is necessary to develop new numerical methods for solving such problems. So, we propose a new boundary element model for the solution of general and complex problems associated with this theory. The numerical results are presented graphically in order to display the effect of the graded parameter on the temperatures and displacements. The numerical results also confirm the validity and accuracy of our proposed model.

1. Introduction

Functionally graded material (FGM) is a special type of advanced inhomogeneous materials. Functionally graded structure is a mixture of two or more distinct materials (usually heat-resisting ceramic on the outside surface and fracture-resisting metal on the inside surface) that have specified properties in specified direction of the structure to achieve a require function [1, 2]. This feature enables obtaining structures with the best of both material’s properties, and suitable for applications requiring high thermal resistance and high mechanical strength [3, 4, 5, 6, 7, 8, 9, 10, 11, 12].

Functionally Graded Materials have been wide range of thermoelastic applications in several fields, for example, the water-cooling model of a fusion reactor divertor is one of the most widely used models in industrial design, which is consisting of a tungsten (W) and a copper (Cu), that subjected to a structural integrity issue due to thermal stresses resulted from thermal expansion mismatch between the bond materials. Recently, functionally graded tungsten (W)–copper (Cu) has been developed by using a precipitation-hardened copper alloy as matrix instead of pure copper, to overcome the loss of strength due to the softening of the copper matrix.

The carbon nanotubes (CNT) in FGM have new applications such as reinforced functionally graded piezoelectric actuators, reinforced functionally graded polyestercalcium phosphate materials for bone replacement, reinforced functionally graded tools and dies for reduce scrap, better wear resistance, better thermal management, and improved process productivity, reinforced metal matrix functionally graded composites used in mining, geothermal drilling, cutting tools, drills and machining of wear resistant materials. Also, they used as furnace liners and thermal shielding elements in microelectronics.

There are many areas of application for elastic and thermoelastic functionally graded materials, for example, industrial applications such as MRI scanner cryogenic tubes, eyeglass frames, musical instruments, pressure vessels, fuel tanks, cutting tool inserts, laptop cases, wind turbine blades, firefighting air bottles, drilling motor shaft, X-ray tables, helmets and aircraft structures. Automobiles applications such as combustion chambers, engine cylinder liners, leaf springs, diesel engine pistons, shock absorbers, flywheels, drive shafts and racing car brakes. Aerospace applications rocket nozzle, heat exchange panels, spacecraft truss structure, reflectors, solar panels, camera housing, turbine wheels and Space shuttle. Submarine applications such as propulsion shaft, cylindrical pressure hull, sonar domes, diving cylinders and composite piping system. Biotechnology applications such as functional gradient nanohydroxyapatite reinforced polyvinyl alcohol gel biocomposites. Defense applications such as armor plates and bullet-proof vests. High-temperature environment applications such as aerospace and space vehicles. Biomedical applications such as orthopedic applications for teeth and bone replacement. Energy applications such as energy conversion devices and as thermoelectric converter for energy conservation. They also provide thermal barrier and are used as protective coating on turbine blades in gas turbine engine. Marine applications such as parallelogram slabs in buildings and bridges, swept wings of aircrafts and ship hulls. Optoelectronic applications such as automobile engine components, cutting tool insert coating, nuclear reactor components, turbine blade, tribology, sensors, heat exchanger, fire retardant doors, etc.

According to continuous and smooth variation of FGM properties throughout in depth, there are many laws to describe the behavior of FGM such as index [13], sigmoid law [14], exponential law [15] and power law [16, 17, 18, 19, 20, 21, 22, 23, 24].

There was widespread interest in functionally graded materials, which has developed a lot of analytical methods for analysis of elasticity [25, 26, 27, 28, 29, 30, 31, 32] and thermoelasticity [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53] problems, some of which have become dominant in scientific literature. For the numerical methods, the isogeometric finite element method (FEM) has been used by Valizadeh et al. [54] for static characteristics of FGM and by Bhardwaj et al. [55] for solving crack problem of FGM. Nowadays, the boundary element method is a simple, efficient and powerful numerical tool which provides an excellent alternative to the finite element method for the solution of FGM problems, Sladek et al. [56, 57, 58] have been developed BEM formulation for transient thermal problems in FGMs. Gao et al. [59] developed fracture analysis of functionally graded materials by a BEM. Fahmy [60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72] developed BEM to solve elastic, thermoelastic and biomechanic problems in anisotropic functionally graded structures. Further details on the BEM are given in [73, 74] and the references therein.

In the present paper, we propose new FGA structures theory and new boundary element technique for modeling problems of nonlinear uncoupled magneto-thermoelasticity involving three temperatures. The boundary element method reduces the dimension of the problem, therefore, we obtain a reduction of numerical approximation, linear equations system and input data. Since there is strong nonlinearity in the proposed theory and its related problems. So, we develop new boundary element technique for modeling such problems. The numerical results are presented graphically through the thickness of the homogeneous and functionally graded structures to show the effect of graded parameter on the temperatures and displacements. The numerical results demonstrate the validity and accuracy of our proposed model.

A brief summary of the chapter is as follows: Section 1 outlines the background and provides the readers with the necessary information to books and articles for a better understanding of mechanical behaviour of magneto-thermoelastic FGA structures and their applications. Section 2 describes the formulation of the new theory and its related problems. Section 3 discusses the implementation of the new BEM for solving the nonlinear radiative heat conduction equation, to obtain the three temperature fields. Section 4 studies the development of new BEM and its implementation for solving the move equation based on the known three temperature fields, to obtain the displacement field. Section 5 presents the new numerical results that describe the through-thickness mechanical behaviour of homogeneous and functionally graded structures.

2. Formulation of the problem

We consider a Cartesian coordinate system for 2D structure (see Figure 1) which is functionally graded along the $0\mathrm{x}$ direction, and considering $\mathrm{z}$ -axis is the direction of the effect of the constant magnetic field ${\mathrm{H}}_0$ .

Figure 1.

Geometry of the FGA structure.

The fractional-order governing equations of three temperatures nonlinear uncoupled magneto-thermoelasticity in FGA structures can be written as follows [6].

E1

E2

E3

where

,

, $u_k$ , $C_{\mathit{\text{pjkl}}}$ ( $C_{\mathit{\text{pjkl}}}=C_{\mathit{\text{klpj}}}=C_{\mathit{\text{kljp}}}$ ),

(

), μ and $h_p$ are respectively, mechanic stress tensor, Maxwell stress tensor, displacement, constant elastic moduli, stress-temperature coefficients, magnetic permeability and perturbed magnetic field.

The nonlinear time-dependent two dimensions three temperature (2D-3 T) radiation diffusion equations coupled by electron, ion and phonon temperatures may be written as follows

E4

where

E5

and

E6

The total energy per unit mass can be expressed as follows

E7

where

are conductive coefficients,

are temperature functions,

are isochore specific-heat coefficients, ρ is the density, τ is the time. In which,

,

, $\mathbb{B}$ , ${\mathbb{A}}_{\mathit{ei}}$ , ${\mathbb{A}}_{\mathit{ep}}$ are constant inside each subdomain, ${\mathbb{W}}_{\mathit{ei}}$ and ${\mathbb{W}}_{\mathit{ep}}$ are electron-ion coefficient and electron–phonon coefficient, respectively.

Initial and boundary conditions can be written as

E8

E9

E10

E11

E12

E13

E14

E15

3. BEM numerical implementation for temperature field

This section outlines the solution of 2D nonlinear time-dependent three temperatures (electron, ion and phonon) radiation diffusion equations using a boundary element method.

Now, let us consider

and discretize the time interval $\left[0,F\right]$ into $F+1$ equal time steps, where

,

Let

be the solution at time

. Assuming that the time derivative of temperature within the time interval

can be approximated by.

E16

denotes the Caputo fractional time derivative of order $a$ defined by [75].

E17

By using a finite difference scheme of Caputo fractional time derivative of order $a$ (17) at times

and

, we obtain:

E18

Where

E19

E20

According to Eq. (18), the fractional order heat Eq. (4) can be replaced by the following system

E21

According to Fahmy [60], and using the fundamental solution which satisfies the system (21), the boundary integral equations corresponding to nonlinear three temperature heat conduction-radiation equations can be written as

E22

which can be written in the absence of internal heat sources as follows

E23

Time temperature derivative can be written as

E24

where $f^j\left(r\right)$ are known functions and

are unknown coefficients.

We suppose that

is a solution of

E25

Then, Eq. (23) yields the following boundary integral equation

E26

where

E27

and

E28

In which the entries of $f_{\mathit{ji}}^{-1}$ are the coefficients of $F^{-1}$ with matrix $F$ defined as [76].

E29

Using the standard boundary element discretization scheme for Eq. (26) and using Eq. (28), we have

E30

The diffusion matrix can be defined as

E31

with

$${\left\{\widehat{\mathrm{T}}\right\}}_{\mathrm{ij}}={\widehat{\mathrm{T}}}^{\mathrm{j}}\left({\mathrm{x}}_{\mathrm{i}}\right)$$ E32

$${\left\{\widehat{\mathrm{Q}}\right\}}_{\mathrm{ij}}={\widehat{\mathrm{q}}}^{\mathrm{j}}\left({\mathrm{x}}_{\mathrm{i}}\right)$$ E33

In order to solve Eq. (30) numerically the functions

and $q$ are interpolated as

E34

E35

where

determines the practical time τ in the current time step.

By differentiating Eq. (34) with respect to time we get

E36

The substitution of Eqs. (34)–(36) into Eq. (30) leads to

E37

By using initial and boundary conditions, we get

E38

This system yields the temperature, that can be used to solve (1) for the displacement.

4. BEM numerical implementation for displacement field

Based on Eqs. (2) and (3), Eq. (1) can be rewritten as

E39

where

E40

when the temperatures are known, the displacement can be computed by solving (39) using BEM. By choosing $u_p^{\ast }$ as the weight function and applying the weighted residual method, Eq. (39) can be reexpressed as

E41

The first term in (41) can be integrated partially using Gau β theory yields

$${\int }_{\mathrm{R}}{C}_{\mathit{\text{pjkl}}}{u}_{k,\mathit{lj}}{u}_p^{\ast }d\mathrm{R}={\int }_{\mathrm{C}}{C}_{\mathit{\text{pjkl}}}{u}_{k,l}{u}_p^{\ast }{n}_{j}d\mathrm{C}-{\int }_{\mathrm{R}}{C}_{\mathit{\text{pjkl}}}{u}_{k,l}{u}_{p,j}^{\ast }d\mathrm{R}$$ E42

The last term in (42) can be integrated partially twice using Gau β theory yields

$${\int }_{\mathrm{R}}{C}_{\mathit{\text{pjkl}}}{u}_{k,l}{u}_{p,j}^{\ast }d\mathrm{R}={\int }_{\mathrm{C}}{C}_{\mathit{\text{pjkl}}}{u}_k{u}_{p,j}^{\ast }{n}_{l}d\mathrm{C}-{\int }_{\mathrm{R}}{C}_{\mathit{\text{pjkl}}}{u}_k{u}_{p,\mathit{jl}}^{\ast }d\mathrm{R}$$ E43

Based on Eq. (43), Eq. (42) can be rewritten as

$${\int }_{\mathrm{R}}{C}_{\mathit{\text{pjkl}}}{u}_{k,\mathit{lj}}{u}_p^{\ast }d\mathrm{R}-{\int }_{\mathrm{R}}{C}_{\mathit{\text{pjkl}}}{u}_k{u}_{p,\mathit{jl}}^{\ast }d\mathrm{R}={\int }_{\mathrm{C}}{C}_{\mathit{\text{pjkl}}}{u}_{k,l}{u}_p^{\ast }{n}_{j}d\mathrm{C}-{\int }_{\mathrm{C}}{C}_{\mathit{\text{pjkl}}}{u}_k{u}_{p,j}^{\ast }{n}_{l}d\mathrm{C}$$ E44

which can be written as

E45

The boundary tractions are

$$t_p=C_{\mathit{\text{pjkl}}}{u}_{k,l}{n}_j=G_{\mathit{jl}}{u}_k\text{and}{t}_p^{\ast }=C_{\mathit{\text{pjkl}}}{u}_{k,j}^{\ast }{n}_l=G_{\mathit{jl}}^{\ast }{u}_k^{\ast }$$ E46

By using the symmetry relation of elasticity tensor, we obtain

E47

E48

Using Eqs. (46)–(48), the Eq. (45) can be reexpressed as

E49

We define the fundamental solution $u_{\mathit{mk}}^{\ast }$ by the relation

E50

By modifying the weighting functions, Eq. (49) can be written as

E51

From (39), (50) and (51), the representation formula may be written as

E52

Let

E53

The displacement particular solution may be defined as

E54

Differentiation of (54) leads to.

E55

Now, we obtain the traction particular solution $t_{\mathit{pn}}^q$ and source function $f_{\mathit{pn}}^q$ as

$${\mathrm{t}}_{\mathrm{pn}}^{\mathrm{q}}={\mathrm{C}}_{\text{pjkl}}{\mathrm{u}}_{\mathrm{kn},\mathrm{l}}^{\mathrm{q}}{\mathrm{n}}_{\mathrm{j}},{\mathrm{L}}_{\mathrm{jl}}{\mathrm{u}}_{\mathrm{kn}}^{\mathrm{q}}={\mathrm{f}}_{\mathrm{pn}}^{\mathrm{q}}$$ E56

The domain integral may be approximated as follows

E57

The use of (57) together with the dual reciprocity

$${\int }_{\mathrm{R}}\left(L_{\mathit{jl}}{u}_{\mathit{kn}}^q{u}_{\mathit{mp}}^{\ast }-L_{\mathit{jl}}{u}_{\mathit{mk}}^{\ast }{u}_{\mathit{pn}}^q\right)d\mathrm{R}={\int }_{\mathrm{C}}\left(u_{\mathit{mp}}^{\ast }{t}_{\mathit{pn}}^q-t_{\mathit{mp}}^{\ast }{u}_{\mathit{pn}}^q\right)d\mathrm{C}$$ E58

Leads to

E59

From (50), we can write

E60

By using (52), (59) and (60), we obtain

E61

According to Fahmy [9, 10, 11], the right-hand side integrals of (61) can be reexpressed as

E62

and

E63

According to Fahmy [12], Guiggiani and Gigante [77] and Mantič [78] Eqs. (62) and (63) can respectively be expressed as

E64

E65

By using (64) and (65), the dual reciprocity boundary integral equation becomes

E66

On the basis of isoparametric concept, we can write

E67

E68

By implementing the point collocation procedure and using (67) and (68), Eq. (66) may be reexpressed as

E69

Let us suppose that

E70

E71

E72

We can write (69) as follows

E73

By using the point collocation procedure,

can be calculated from (53) as

E74

Now, from (74), we may derive

E75

From (73) using (75) we have

E76

where

E77

By considering the following known $k$ and unknown $u$ superscripts nodal vectors

E78

Hence (76) may be written as

E79

From the first row of (79), we can calculate the unknown fluxes

as follows

E80

From the second row of (79) and using (80) we get

E81

where

E82

Eq. (81) can be written at $\left(n+1\right)$ time step as

E83

where

E84

In order to solve (83), The implicit backward finite difference scheme has been applied based on the Houbolt’s algorithm and the following approximations

E85

E86

By using (85) and (86), we have from (83)

E87

In which.

E88

E89

We implement the successive over-relaxation (SOR) of Golub and Van Loan [79] for solving (87) to obtain

. Then, the unknown

and

can be obtained from (76) and (77), respectively. By using the procedure of Bathe [80], we obtain the traction vector $t_{n+1}^u$ from (73).

5. Numerical results and discussion

The BEM that has been used in the current paper can be applicable to a wide variety of FGA structures problems associated with the proposed theory of three temperatures nonlinear uncoupled magneto-thermoelasticity. In order to evaluate the influence of graded parameter on the three temperatures and displacements, the numerical results are carried out and depicted graphically for homogeneous ( $m=0$ ) and functionally graded ( $m=0.5\text{and}1.0$ ) structures.

Figures 2–4 show the distributions of the three temperatures $T_e,T_i\text{and}{T}_p$ through the thickness coordinate $\mathit{Ox}$ . It was shown from these figures that the three temperatures increase with increasing value of graded parameter $\mathrm{m}$ .

Figure 2.

Variation of the electron temperature T_e through the thickness coordinate x.

Figure 3.

Variation of the ion temperature T_i through the thickness coordinate x.

Figure 4.

Variation of the photon temperature T_p through the thickness coordinate x.

Figures 5 and 6 show the distributions of the displacements ${\mathrm{u}}_1$ and ${\mathrm{u}}_2$ through the thickness coordinate $\mathrm{Ox}$ . It was noticed from these figures that the displacement components increase with increasing value of graded parameter $\mathrm{m}$ .

Figure 5.

Variation of the displacement u₁ through the thickness coordinate x.

Figure 6.

Variation of the displacement u₂ through the thickness coordinate x.

Figures 7 and 8 show the distributions of the displacements $u_1$ and $u_2$ with the time for boundary element method (BEM), finite difference method (FDM) and finite element method (FEM) to demonstrate the validity and accuracy of our proposed technique. It is noted from numerical results that the BEM obtained results are agree quite well with those obtained using the FDM of Pazera and Jędrysiak [81] and FEM of Xiong and Tian [82] results based on replacing heat conduction with three-temperature heat conduction.

Figure 7.

Variation of the displacement u₁ with time τ.

Figure 8.

Variation of the displacement u₂ with time τ.

6. Conclusion

The main aim of this article is to introduce a new fractional-order theory called nonlinear uncoupled magneto-thermoelasticity theory involving three temperatures for FGA structures and new boundary element technique for solving problems related to the proposed theory. Since the nonlinear three temperatures radiative heat conduction equation is independent of the displacement field, we first determine the temperature field using the BEM, then based on the known temperature field, the displacement field is obtained by solving the move equation using the BEM. It can be seen from the numerical results that the graded parameter had a significant effect on the temperatures and displacements through the thickness of the functionally graded structures. Since there are no available results for the considered problem. So, some literatures may be considered as special cases from the considered problem based on replacing the heat conduction by three temperatures radiative heat conduction. The numerical results demonstrate the validity and accuracy of our proposed model. From the proposed BEM technique that has been performed in the present paper, it is possible to conclude that the proposed BEM should be applicable to any FGM uncoupled magneto-thermoelastic problem with three-temperature. BEM is more efficient, accurate and easy to use than FDM or FEM, because it only needs to solve the unknowns on the boundaries and BEM users need only to deal with real geometry boundaries. Also, BEM is reducing the computational cost of its solver. The present numerical results for our complex problem may provide interesting information for computer scientists, designers of new FGM materials and researchers in FGM science as well as for those working on the development of new functionally graded structures.